S.O.S. Mathematics CyberBoard

LJB · Posted: Thu, 9 Feb 2006 20:02:41 UTC Post subject: not locally connected

The question reads:
Show that {1/n : n in N} u {0} with the standard topology induced from R is not locally connected.

Definitions I know:
A topological space (X,T) is called locally connected if for each x in X, and any open set U containing x, there is a connected open set V in U such that x is in V.

Things I know:
The integers N with the standard topology induced from R is locally connected.
If there exists a point s.t. if you take any ball around it, it is not connected shows that the space is not locally connected

The problem: I don't know which point to choose and equally am not sure how to show that it is not connected. The graph appears to be connected at all N and so any suggestions would be appreciated.

Thank you for your help,

LJB

Bilbo · Posted: Thu, 9 Feb 2006 21:05:17 UTC Post subject: Re: not locally connected

LJB · Posted: Thu, 9 Feb 2006 21:39:27 UTC Post subject: Disjoint union of open sets

Thank you for your help.

Let any ball around 0 be the open ball, B_epsilon (0) where epsilon > 0

Is this correct?

Also, does that mean that B_epsilon(0) is the disjoint union of B_delta({1/n: n in N}) where 0 < epsilon < delta?

Thank you,

LJB

Bilbo · Posted: Thu, 9 Feb 2006 23:28:15 UTC Post subject:

Not quite.

I would write it like this:

for

small enough and this ball is disjoint from the rest of the space. So

can be written as a disjoint union

for suitable

.

You're welcome.

LJB · Posted: Thu, 9 Feb 2006 23:49:30 UTC Post subject: An extension

Thank you very much for your help. That makes a lot of sense.

Out of curiosity, is there a connected component of that space which isn't open?

My reason for the question is as follows:
Just because it isn't locally connected doesnt mean it can't be connected (I think!) And if it can be connected, then it is usually open, so i was wondering whether there is such an example.

Thank you again,

LJB

Bilbo · Posted: Fri, 10 Feb 2006 03:14:16 UTC Post subject: Re: An extension

LJB · Posted: Sun, 12 Feb 2006 12:22:28 UTC Post subject: Alternative

Hi,

I have been having a think about the question again, and have come up with a second proof, but was wondering whether it works!!

Consider {0} and for any epsilon > 0 B_epsilon (0)
Take delta in R \ {1/n : n in N} u {0} s.t. delta is in B_epsilon (0)
Partition B_epsilon (0) into A = (0, delta), B = (delta, epsilon) which partition B_epsilon (0) = (0,epsilon)
All the balls are disjoint and so {1/n : n in N} u {0} is not locally connected.

If anyone has any suggestions for making the last line more thorough I would appreciate it!

Thank you,

LJB

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